House Speaker John A. Boehner will head to the White House on Friday to open formal negotiations with President Obama on a deal to avoid the year-end “fiscal cliff” with a new, conciliatory tone.

But Boehner (R-Ohio) still has the same problem that has dogged him since the day he became speaker two years ago: a caucus that does not necessarily believe in the kind of grand compromise a deal will require.

This time around, the chief problem is whether and how to bring in new revenue as part of a deal to avoid $500 billion in tax increases and spending cuts due to start in January. Boehner has indicated a fresh willingness to include new tax money in an agreement, but how far he wants to go is unclear even to his own wary members.

Many Republican lawmakers, meanwhile, continue to say that projected revenue from economic growth is the only kind of higher tax collection they will support.

“If both sides agree that we can get more revenue through economic growth, then we can agree,” said Rep. Tim Huelskamp (R-Kan.). But he said any proposal brokered by Boehner that would bring in new money through higher tax burdens would “split the party wide open.”

Boehner stands atop a nearly unchanged caucus of conservatives who have spent their first week back in Washington since the Nov. 6 election bucking one another up as the town’s last check on Democratic power. Rather than feeling chastened by the Democratic gains, many House conservatives say they feel empowered by Americans’ renewed endorsement of divided government.

Their hard-line approach could complicate Boehner’s effort to reach a deal with the White House, particularly with Democrats now equally determined to use their electoral victories to ensure higher taxes on the rich as part of a deficit-reduction deal.

Already, Obama has opened talks indicating that he would like to see $1.6 trillion in new revenue — double what Boehner had offered in secret debt-reduction talks with the president in the summer of 2011.

At a Capitol Hill event called “Conservations With Conservatives” this week, a group of House members repeatedly scoffed at the notion that Obama’s win, with just over 50 percent of the popular vote, gives him a mandate on the tax issue.

“He has a mandate to talk about it,” said Rep. Raul R. Labrador (R-Idaho). “And we have a mandate to fight it. We will continue to fight any members of our conference who believe that this is the time to raise taxes.”

In private and public pep talks, Boehner has encouraged his members not to relinquish their role as a limit on Obama’s second-term agenda.

“Our majority is a primary line of defense for the American people against a government that spends too much, borrows too much when left unchecked,” he said Wednesday, shortly after his colleagues unanimously reelected him as speaker.

Many Democrats insist that a deal must allow President George W. Bush-era cuts in tax rates to expire on schedule next month for those making a net income of more than $250,000 a year. That would mean allowing the top marginal income tax rate to rise from 35 percent to 39.6 percent, where it stood under President Bill Clinton.

At the same time, Boehner has said he is ready to compromise on revenue — indicating that he could accept more tax money flowing into the nation’s coffers if it was collected in ways other than raising rates, such as by limiting deductions or closing loopholes.

Democrats say the Republicans are engaging in magical thinking — predicting new money that may never show up even as the debt grows. Although Obama said Wednesday that he would not draw “red lines” on the 39.6 percent rate, he also indicated he will not accept a deal that does not include a guarantee of more taxes paid by the wealthy.

Many Republicans in the Senate are more realistic about the parameters of a bipartisan agreement. At Wednesday’s GOP Senate policy lunch, several offered ideas for elements of a potential bargain, including Patrick J. Toomey (Pa.) and and Bob Corker (Tenn.). Neither would provide details, saying they want to let talks between Obama and Boehner take their course. But both said they are willing to raise the tax burden in exchange for significant entitlement reform. Two people familiar with Corker’s plan said it would raise about $800 billion in the next decade by imposing a $50,000 limit on deductions.

“We’re not going to expect the Democrats to say economic growth is revenue. That’s not going to work. We’ve got to get real revenue,” said Sen. Lindsey O. Graham (R-S.C.). “If Boehner and Obama can come together with a deal that sails through the House, start in the House. But if you can’t get a deal from Boehner and Obama, [it might make sense to] start it in the Senate.”

By being publicly vague on the issue, Boehner provides himself additional flexibility in talks with Obama. But he may not be preparing House members for the kind of deal they may be asked to support to avoid flying off the cliff, an outcome predicted to plunge the economy back into a recession.

One potential wild card in the GOP caucus might be at least 29 Republicans who are retiring or lost their elections and could provide support for a deal.

“The vast majority of Americans think that is the way to get this country back on track,” said Rep. Charles F. Bass (R-N.H.), a more moderate member who lost his election last week and said he is prepared to accept higher marginal tax rates on the wealthy, provided a deal also included major spending cuts.

Rep. Nan A.S. Hayworth (R-N.Y.) said her defeat had done nothing to change her belief that higher taxes are a bad idea. But, she added: “We’ve had an election in which there was a Senate majority that was chosen, for whatever reason, and the president was reelected. And the president has been adamant on raising taxes. So some form of a tax increase will probably occur on the wealthiest — quote, unquote — will occur.”

Boehner’s position has been bolstered by apparent unity with House Majority Leader Eric Cantor (R-Va.), who enjoys special trust with the caucus’s most conservative members, as well as former vice-presidential candidate Paul Ryan (R-Wis.), chairman of the Budget Committee.

Ryan is one of three committee chairman whom Boehner has asked to start attending a daily morning meeting of House leaders, which could serve to keep him more closely involved with negotiations.

Boehner has told fellow Republicans not to expect negotiations to begin to bear fruit until after Thanksgiving. Then he will face a now-familiar task — persuading some of his most conservative members that a deal does not represent a capitulation, or finding votes for a deal without them.

“Compromise has a very small constituency,” said Rep. Trey Gowdy (R-S.C.). “Very small.”

Lori Montgomery contributed to this report.